Chapter 3 - Collinearity and Concurrence

Summary

But he opened out the hinges, Pushed and pulled the joints and hinges, Till it looked all squares and oblongs Like a complicated figure In the Second Book of Euclid.

C. L. Dodgson

After discussing some further properties of triangles and quadrangles (or quadrilaterals), we shall approach the domain of projective geometry (and even trespass a bit). A systematic development of that fascinating subject must be left for another book, but four of its most basic theorems are justifiably mentioned here because they can be proved by the methods of Euclid; in fact, three of the four are so old that no other methods were available at the time of their discovery. All these theorems deal either with collinearity (certain sets of points lying on a line) or concurrence (certain sets of lines passing through a point). The spirit of projective geometry begins to emerge as soon as we notice that, for many purposes, parallel lines behave like concurrent lines.

Quadrangles; Varignon's theorem

A polygon may be defined as consisting of a number of points (called vertices) and an equal number of line segments (called sides), namely a cyclically ordered set of points in a plane, with no three successive points collinear, together with the line segments joining consecutive pairs of the points. In other words, a polygon is a closed broken Sine lying in a plane.